Best Known (144−40, 144, s)-Nets in Base 5
(144−40, 144, 408)-Net over F5 — Constructive and digital
Digital (104, 144, 408)-net over F5, using
- 4 times m-reduction [i] based on digital (104, 148, 408)-net over F5, using
- trace code for nets [i] based on digital (30, 74, 204)-net over F25, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 30 and N(F) ≥ 204, using
- net from sequence [i] based on digital (30, 203)-sequence over F25, using
- trace code for nets [i] based on digital (30, 74, 204)-net over F25, using
(144−40, 144, 1486)-Net over F5 — Digital
Digital (104, 144, 1486)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5144, 1486, F5, 40) (dual of [1486, 1342, 41]-code), using
- 1341 step Varšamov–Edel lengthening with (ri) = (9, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 31 times 0, 1, 33 times 0, 1, 34 times 0, 1, 36 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 43 times 0, 1, 44 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 52 times 0, 1, 55 times 0, 1, 58 times 0) [i] based on linear OA(540, 41, F5, 40) (dual of [41, 1, 41]-code or 41-arc in PG(39,5)), using
- dual of repetition code with length 41 [i]
- 1341 step Varšamov–Edel lengthening with (ri) = (9, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 31 times 0, 1, 33 times 0, 1, 34 times 0, 1, 36 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 43 times 0, 1, 44 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 52 times 0, 1, 55 times 0, 1, 58 times 0) [i] based on linear OA(540, 41, F5, 40) (dual of [41, 1, 41]-code or 41-arc in PG(39,5)), using
(144−40, 144, 223770)-Net in Base 5 — Upper bound on s
There is no (104, 144, 223771)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 44845 003619 532041 144061 855527 245611 358713 600352 350399 530502 342703 155168 251598 197023 683360 647274 596401 > 5144 [i]